IDEAS home Printed from https://ideas.repec.org/a/spr/jogath/v53y2024i2d10.1007_s00182-023-00881-0.html
   My bibliography  Save this article

Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game

Author

Listed:
  • Arvind Srinivasan

    (University of Colorado Denver)

  • Burton Simon

    (University of Colorado Denver)

Abstract

The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, $$(p(1),p(2),\ldots )$$ ( p ( 1 ) , p ( 2 ) , … ) , where p(k) is the probability you pick k. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, n, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large n, players should choose a number uniformly between 1 and $$\phi _n \sim O(n/\ln (n))$$ ϕ n ∼ O ( n / ln ( n ) ) . Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for $$\phi _n$$ ϕ n , and efficient ways to sample from the continuous approximation.

Suggested Citation

  • Arvind Srinivasan & Burton Simon, 2024. "Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 653-671, June.
  • Handle: RePEc:spr:jogath:v:53:y:2024:i:2:d:10.1007_s00182-023-00881-0
    DOI: 10.1007/s00182-023-00881-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00182-023-00881-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s00182-023-00881-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Simone Pigolotti & Sebastian Bernhardsson & Jeppe Juul & Gorm Galster & Pierpaolo Vivo, 2011. "Equilibrium strategy and population-size effects in lowest unique bid auctions," Papers 1105.0819, arXiv.org, revised Feb 2012.
    2. Roger B. Myerson, 1998. "Population uncertainty and Poisson games," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 375-392.
    3. Robert Östling & Joseph Tao-yi Wang & Eileen Y. Chou & Colin F. Camerer, 2011. "Testing Game Theory in the Field: Swedish LUPI Lottery Games," American Economic Journal: Microeconomics, American Economic Association, vol. 3(3), pages 1-33, August.
    4. Simone Pigolotti & Sebastian Bernhardsson & Jeppe Juul & Gorm Galster & Pierpaolo Vivo, 2012. "Equilibrium strategy and population-size effects in lowest unique bid auctions," Post-Print hal-00681002, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Berger, Ulrich & De Silva, Hannelore & Fellner-Röhling, Gerlinde, 2016. "Cognitive hierarchies in the minimizer game," Journal of Economic Behavior & Organization, Elsevier, vol. 130(C), pages 337-348.
    2. Mohlin, Erik & Östling, Robert & Wang, Joseph Tao-yi, 2015. "Lowest unique bid auctions with population uncertainty," Economics Letters, Elsevier, vol. 134(C), pages 53-57.
    3. Mohlin, Erik & Östling, Robert & Wang, Joseph Tao-yi, 2020. "Learning by similarity-weighted imitation in winner-takes-all games," Games and Economic Behavior, Elsevier, vol. 120(C), pages 225-245.
    4. Yamada, Takashi & Hanaki, Nobuyuki, 2016. "An experiment on Lowest Unique Integer Games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 88-102.
    5. Cancan Zhou & Hongguang Dong & Rui Hu & Qinghua Chen, 2015. "Smarter than Others? Conjectures in Lowest Unique Bid Auctions," PLOS ONE, Public Library of Science, vol. 10(4), pages 1-13, April.
    6. Rui Hu & Jinzhong Guo & Qinghua Chen & Tao Zheng, 2017. "The Psychological Force Model for Lowest Unique Bid Auction," Computational Economics, Springer;Society for Computational Economics, vol. 50(4), pages 655-667, December.
    7. Erik Mohlin & Robert Ostling & Joseph Tao-yi Wang, 2014. "Learning by Imitation in Games: Theory, Field, and Laboratory," Economics Series Working Papers 734, University of Oxford, Department of Economics.
    8. Pierre Bernhard & Marc Deschamps, 2017. "On Dynamic Games with Randomly Arriving Players," Dynamic Games and Applications, Springer, vol. 7(3), pages 360-385, September.
    9. Yuhong Xu & Shih-Fen Cheng & Xinyu Chen, 2023. "Improving Quantal Cognitive Hierarchy Model Through Iterative Population Learning," Papers 2302.06033, arXiv.org, revised Feb 2023.
    10. Camerer, Colin F. & Ho, Teck-Hua, 2015. "Behavioral Game Theory Experiments and Modeling," Handbook of Game Theory with Economic Applications,, Elsevier.
    11. Nadir Altinok & Abdurrahman Aydemir, 2015. "The Unfolding of Gender Gap in Education," Working Papers 934, Economic Research Forum, revised Aug 2015.
    12. Costa-Gomes, Miguel A. & Shimoji, Makoto, 2014. "Theoretical approaches to lowest unique bid auctions," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 16-24.
    13. Yusuke Matsuki, 2016. "A Distribution-Free Test of Monotonicity with an Application to Auctions," Working Papers e110, Tokyo Center for Economic Research.
    14. Le Breton, Michel & Lepelley, Dominique & Smaoui, Hatem, 2012. "The Probability of Casting a Decisive Vote: From IC to IAC trhough Ehrhart's Polynomials and Strong Mixing," IDEI Working Papers 722, Institut d'Économie Industrielle (IDEI), Toulouse.
    15. Micael Castanheira, 2003. "Why Vote For Losers?," Journal of the European Economic Association, MIT Press, vol. 1(5), pages 1207-1238, September.
    16. Ming Li & Dipjyoti Majumdar, 2010. "A Psychologically Based Model of Voter Turnout," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 12(5), pages 979-1002, October.
    17. Alastair Smith & Bruce Bueno de Mesquita & Tom LaGatta, 2017. "Group incentives and rational voting1," Journal of Theoretical Politics, , vol. 29(2), pages 299-326, April.
    18. Andonie, Costel & Kuzmics, Christoph, 2012. "Pre-election polls as strategic coordination devices," Journal of Economic Behavior & Organization, Elsevier, vol. 84(2), pages 681-700.
    19. François Durand & Antonin Macé & Matias Nunez, 2019. "Analysis of Approval Voting in Poisson Games," Working Papers halshs-02049865, HAL.
    20. Pierre Bernhard & Marc Deschamps, 2016. "Dynamic equilibrium in games with randomly arriving players," Working Papers 2016-10, CRESE.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jogath:v:53:y:2024:i:2:d:10.1007_s00182-023-00881-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.